The syntax of SL
Talking about a language
When we use English to discuss SL, English is the metalanguage and SL is the object language. To mention an expression rather than use it, the book puts it in single quotes or displays it on its own line: Minnesota is a state, but 'Minnesota' is a word.
Bold capitals P, Q, R, S are metavariables. They belong to English and stand for any sentence of SL, the way x stands for any number. They are not sentences of SL themselves.
Vocabulary
Sentence letters are the capitals A to Z, with or without positive-integer subscripts (A, B1, C2…), so there are infinitely many. The tilde is a unary connective; the other four are binary. Parentheses are the only punctuation.
Atomic and compound
Sentence letters are the atomic sentences. Everything else is compound and has one main connective plus one or two immediate components. The components of a sentence are the sentence itself, its immediate components, their immediate components, and so on down to the letters.
Names to know: the immediate components of a conjunction are its conjuncts, of a disjunction its disjuncts, and of a material conditional its antecedent (before the horseshoe) and consequent (after it).
Two conventions
- Outermost parentheses may be dropped: A ⊃ (B & ~ D). This never applies to a negation, which starts with a tilde, not a parenthesis.
- Square brackets may replace parentheses to make long sentences readable.
Recursive definition of "sentence of SL"
- Every sentence letter is a sentence.
- If P is a sentence, so is ~ P.
- If P and Q are sentences, so is (P & Q).
- … so is (P ∨ Q).
- … so is (P ⊃ Q).
- … so is (P ≡ Q).
- Nothing else is a sentence (the closure clause).
Clauses 3–6 always add parentheses. That's why A & B & C isn't a sentence: SL never guesses the grouping.
Sentence checker
Type any string of symbols. The checker applies the recursive definition: it says whether the string is a sentence of SL, names its main connective, lists every component, and shows how the sentence is built clause by clause.
Introduction to symbolization
Why structure matters
The section opens with the whodunit: either the maid or the butler killed Watson; if it was the maid, Watson was poisoned; he wasn't poisoned; so the butler did it. Give each premise its own letter and the argument looks invalid. Symbolizing the parts (M, B, P) exposes the structure that makes it valid.
Truth-functional use
Two stages
Symbolizing happens in two steps. First write a truth-functional paraphrase in English using only "it is not the case that", "and", "or", "if … then", and "if and only if". Then translate the paraphrase into SL with a symbolization key. In a key each letter stands for exactly one sentence; mnemonic letters help but aren't required.
Patterns from the section
- Negation often has no "not": nonmetal, unmarried. And "Someone isn't telling the truth" negates "Everyone is telling the truth", not "Someone is telling the truth".
- Conjunction covers but, however, although, nevertheless, moreover, while. The contrast these suggest is lost; what's asserted is kept.
- Conditionals: if, provided that, assuming introduce the antecedent. Only if introduces the consequent.
- Biconditionals: if and only if, if but only if, and sometimes just in case.
Characteristic truth-tables
| P | ~ P |
|---|---|
| T | F |
| F | T |
| P | Q | P & Q | P ∨ Q | P ⊃ Q | P ≡ Q |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | F | F |
| F | T | F | T | T | F |
| F | F | F | F | T | T |
A material conditional is false on one row only: true antecedent, false consequent.
"If" versus "only if"
The book's example: a student graduates only if they've taken two semesters of a language. Taking the language doesn't guarantee graduating, so the arrow can't point from it.
More complex symbolizations
The paraphrasing guidelines, in short
- Decide whether the sentence is a truth-functional compound. If not, it is its own paraphrase. If so, check its components the same way.
- Rewrite it with the five standard connectives.
- Use parentheses and brackets to show which sentences each connective joins.
- Put arguments in standard form: premises, a line, then the conclusion.
- Make every component a complete sentence with no cross-references, and pick one wording for each claim.
Work from the outside in
Find the main connective of the English sentence first, paraphrase that, then repeat inside each component. Long sentences become manageable when you only ever deal with one connective at a time.
Cases worth practising
- Neither p nor q: ~ (P ∨ Q), or equivalently ~ P & ~ Q.
- Not both p and q: ~ (P & Q).
- p unless q: P ∨ Q.
- p or q, but not both (exclusive or): (P ∨ Q) & ~ (P & Q).
- At least one of several: a chain of wedges. All or none: a disjunction of two conjunctions.
Summary of common connectives
| English | Symbolization |
|---|---|
| not p | ~ P |
| p and q · p but q · p however q · p although q · p nevertheless q · p nonetheless q · p moreover q | P & Q |
| p or q · p unless q | P ∨ Q |
| p or q (exclusive sense) | (P ∨ Q) & ~ (P & Q) |
| if p then q · p only if q · q if p · q provided that p · q given p | P ⊃ Q |
| p if and only if q · p if but only if q · p just in case q | P ≡ Q |
Adapted from the summary table in section 2.3.
Non-truth-functional uses of connectives
Some English sentences contain connective words but aren't truth-functional compounds: knowing the truth-values of the parts doesn't settle the truth-value of the whole. SL can only treat these as atomic.
Common sources
- Attitudes: believes that, thinks that. Someone can believe a falsehood or fail to believe a truth.
- Likelihood and possibility: it's likely that, it's possible that.
- Causes and reasons: because. Two true sentences joined by "because" can still make a false claim.
- Time order: only after, before.
Conditionals that lose content
Many English conditionals assert a connection that ⊃ can't capture. When the connection is the point, as in a claim based on a scientific law, the material conditional can be true while the English is false. The book's example: someone claims a rod will contract when heated if it's metal; the rod is plastic, so M ⊃ C is true even though the claim is wrong. In cases like that, symbolize the whole sentence as atomic.
The takeaway: results we get later in SL apply to English only as far as our paraphrases capture what the English says.
Quick test
Replace a component with another sentence that has the same truth-value. If the truth-value of the whole could change, the connective isn't being used truth-functionally.
The last one mixes both: the "and" and "isn't" are truth-functional; the belief inside stays atomic.
Truth-table lab
Type a sentence of SL and see its truth-table, one column per compound component. Truth-tables are the subject of Chapter 3; for now this is a handy way to see how each connective behaves and to check a symbolization.
The default sentence is the whodunit from section 2.2, written as one conditional: M (the maid did it), B (the butler did it), P (Watson was poisoned). Shortcuts: ~ or ! for ~, v or | for ∨, > or -> for ⊃, = or <-> for ≡.
Practice sets for Chapter 2
New questions in the style of the 2.1E–2.4E exercises, checked as you go. Do the book's starred exercises too; their answers are in the back.
Discussion questions
- SL demands parentheses around every binary compound. What would go wrong if we allowed A & B ∨ C? Give two English sentences it could stand for.
- Write a sentence that's false as written but true once you add quotation marks. Which side of the use/mention line is each version on?
- Find a real requirement in your own life that fits "p only if q" but not "p if q".
- The book paraphrases "but" as "and". What gets lost, and could that loss ever change whether an argument is valid?
- Is "unless" really inclusive? Think of "I'll walk unless it rains." If it rains and you walk anyway, was the sentence false?
- "Just in case" can mean "if and only if" or "as a precaution". How do you tell which one you're looking at?
- When should a conditional be symbolized as an atomic sentence instead of with ⊃? Bring one example of each.
Questions for our session
Jot down anything that confused you while reading. Saved in this browser only, so bring it up when we meet.